Efficiency of Producing Random Unitary Matrices with Quantum Circuits

We study the scaling of the convergence of several statistical properties of a recently introduced random
unitary circuit ensemble towards their limits given by the circular unitary ensemble. Our study includes the full
distribution of the absolute square of a matrix element, moments of that distribution up to order eight, as well
as correlators containing up to 16 matrix elements in a given column of the unitary matrices. Our numerical
scaling analysis shows that all of these quantities can be reproduced efficiently, with a number of random gates
which scales at most as n_q(\ln n_q /\epsilon)^\nu with the number of qubits n_q for a given fixed precision \epsilon and \nu>0.